Projective range

In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For instance, a correlation interchanges the points of a projective range with the lines of a pencil. A projectivity is said to act from one range to another, though the two ranges may coincide as sets.

A projective range expresses projective invariance of the relation of projective harmonic conjugates. Indeed, three points on a projective line determine a fourth by this relation. Application of a projectivity to this quadruple results in four points likewise in the harmonic relation. Such a quadruple of points is termed a harmonic range. In 1940 Julian Coolidge described this structure and identified its originator:

When a conic is chosen for a projective range, and a particular point E on the comic is selected as origin, then addition of points may be defined as follows:

The circle and hyperbola are instances of a conic and the summation of angles on either can be generated by the method of "sum of points", provided points are associated with angles on the circle and hyperbolic angles on the hyperbola.

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